Talk:Sturm-Liouville theory
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[edit] A bit wordy.
This article is good and easy to read, but too wordy. The distraction about the linear polynomials being eigenfunctions of the Laplacian is unnecessary. All one needs to do is write the formula for the eigenvectors (exp(ikx) where k is even), maybe something like "one checks that exp(ikx), k even, solves the ode."
Similarly, it should be made more clear exactly why it is that arbitrary functions may be written as superpositions of such basic waves. The short answer (applicable to the examples in the article) is Fourier series. The long answer is that the BVP Lu=f subject to Dirichlet boundary data on a nice bounded domain when L is (say) coercive is a map F:f-->u from L^2 to H^1 that is continuous; hence the composition F with the inclusion map i:H^1-->L^2 (which is a compact linear map) gives a compact linear map G=iF, G:L^2-->L^2. This map is self-adjoint if L is self-adjoint, and so it admits a basis of eigenvectors. The eigenvalues of G must accumulate at the origin and nowhere else, hence the eigenvalues of L accumulate at ∞ and nowhere else. This also proves that the set of basic wave functions will necessarely span L^2, even if they are not exp harmonics.
The above is true. I don't see why there needs to be any reference to Fourier Series at all except that sines and cosines form an orthogonal basis under the second differential linear operator because it is a hermitian operator. What should be proven is that L is hermitian (that is that 
) and therefore its eigenfunctions are orthogonal and it's eigenvalues are all real, thus it is possible (as inspiration from Fourier Analysis) to find the coefficients of the inhomogeneous input when you represent it in the basis of those orthogonal eigenfunctions. Accordingly it is then possible to find the coefficients for the solution. For this type of analysis it is important to also note that the hermitian properties of L only hold when u(x) and f(x) are constrained to certain boundary conditions. This could be in the proof that L is, in fact, a hermitian operator.
One should then refer to the Sobolev space article.
I may make the changes some other day if nobody else does it.
Loisel 09:46, 6 Jun 2004 (UTC)
I've rewritten the first half; the second half will come later. Loisel 19:02, 7 Jul 2004 (UTC)
The "normal modes" section is a bit strange. The equation isn't in the Lu=∑D^p(a_{pq}D^qu)+bu=0 form, nor is it in the eigenvalue Lu=λu form. Furthermore, the problem solved in that section is not that of eigenanalysis (for the purpose of solving Lu=f subject to homogeneous boundary conditions) but rather to solve Lu=0 subject to nonhomogeneous boundary conditions.
I think it's a good idea to have a multiple variable example of a S-L problem, and the wave equation is fine with me, but I think the wave operator (what do physicists call it?) and the S-L connection should be made explicit.
Loisel 06:51, 8 Jul 2004 (UTC)
[edit] Rewrite
The introduction to the theory presented here is of little meaning unless boundary conditions are introduced and this should be done at the outset, in the preamble. I'll likely write something up soon.
Mingarelli 11:02, 2 Dec 2005 (EST)
I've just rewritten the opening paragraph here to conform with existing definitions and terminology. For supplementary material see E.L.Ince, Ordinary Differential Equations, Dover Publ., NY (1956).
Mingarelli 11:55, 2 Dec 2005 (EST)
I added more material and precision to the first five or six paragraphs of this article and linked some of the terms with Wikipedia. I'll take a break for now.
Tobias Hein 15:40, 2 Dec 2005 (EST)
I fixed some sign problems in the example of calculating an integrating factor. I need the Sturm- Liouville theory for my diploma thesis in astrophysics on the University of Würzburg (Germany). For more information look at Arfkens "Mathematical Methods for Physicists".
This page has been constantly getting vandalized and renamed "Tony Liuville Theory". I'm too lazy to create an account, but can someone fix this and disable editing?
